Important Dimensionless Numbers

Several important dimensionless numbers in combined heat and momentum transfer in fluids can be derived by considering the simple flow of a Newtonian fluid between two flat plates, one stationary and one moving at velocity, v see Fig. 5.1. 

If it is assumed that only conduction and viscous dissipation play a role of importance, the energy balance can be written as  

When the fluid is Newtonian, the shear stress can be written as (see Section 6.2.1) 9v 

If the pressure gradient in flow direction is assumed to be zero, i.e. only drag flow, the velocity gradient becomes  

This is essentially the same equation as Eq. 7.92 describing the temperature profile in the melt film in the melting region of an extruder. The equation can now be written in the following dimensionless form  

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The most important dimensionless number in fluid mechanics is the Reynolds number (Re), which is defined as... 

In Table 7.7 the most important dimensionless numbers introduced in this chapter are summarized (together with negligibility criteria). With these numbers, it is now possible to give generalized definitions for Aris numbers, for which two situations may be distinguished ... 

This section will present one of the possible physical interpretations of these important dimensionless numbers. First, to show the meaning of Nusselt number, we consider the heat transfer flux in the x direction in the case of a pure molecular mechanism compared with the heat transfer characterizing the process when convection is important. The corresponding fluxes are then written as ... 

Heat-transfer-coefficient correlations are usually presented in terms of dimensionless numbers, which are groups of variables that have no net dimensions when evaluated in any consistent system of units. The most important dimensionless numbers for heat transfer are defined next. 

Table 5. Important dimensionless numbers in neutral transport and reaction in plasma reactors.

Notice that the accumulation rate process and convective forces scale as pV /L, whereas viscous, pressure, and gravity forces scale as pV/L . If one takes the ratio of these two dimensional scaling factors, an important dimensionless number is obtained ... 

The Reynolds number (i.e.. Re) represents an order-of-magnitude ratio of convective forces to viscous forces, and it appears as the most important dimensionless number on the left-hand side of the dimensionless equation of motion ... 

Identify the important dimensionless numbers that appear in the dimensionless equation of continuity for a compressible fluid where the density... 

What important dimensionless number(s) appear in the dimensionless partial differential mass transfer equation for laminar flow through a blood capillary when the important rate processes are axial convection and radial diffusion ... 

This result can be written in terms of the important dimensionless numbers for mass and heat transfer. A completely dimensionless expression is obtained via division of the boundary layer thickness by the cylindrical radius / . If the Reynolds number is defined using R as the characteristic length, instead of the cylindrical diameter, then... 

What is the most important dimensionless number in mass transfer that... 

Identify the important dimensionless number in each case. 

Generally speaking, the fluid behavior can be parameterized using the values taken by some important dimensionless numbers comparing different physical quantities. 

The creation of spherical droplets has been the focus of numerous studies over the years [11-14]. These show conclusively that for both steady and transient flows the onset and mechanisms of droplet breakup can be correlated with the non-dimensional Weber number. It is the most important dimensionless number characterizing droplet formation and can be applied to determine the threshold of droplet formation. However, the critical Weber number is only a sufficient condition for droplet breakup and not a necessary condition. This means that if the critical Weber number is surpassed in a process certainly droplet breakup will occur. But droplet ejection is also possible at lower Weber numbers. The only necessary condition for droplet formation is that the supplied energy is sufficient to overcome friction losses and the surface energy of an ejected droplet. 

The Weber number is the most important dimensionless number characterizing droplet formation. Therefore, the Weber number can be used to distinguish between occurring droplet formation regimes. 

The method of obtaining the important dimensionless numbers from the b sic differential equations is generally the preferred method. In many cases, however, we are not able to formulate a differential equation which clearly applies. Then a more general procedure is required, which is known as the Buckingham method. In this method the listing of the important variables in the particular physical problem is done first. Then we determine the number of dimensionless parameters into which the variables may be combined by using the Buckingham pi theorem. 

Another important dimensionless number is the Nusselt number ... 

Dynamic similarity implies that the important dimensionless numbers must be the same in the model and the prototype. For a particle settling in a fluid. It has been shown that the drag coefficient, Cp, is a function of the dimensionless Reynolds number. Re. By selecting the operating conditions such that Re in the model equals the Re in the prototype, then the drag coefficient (or friction factor) in the prototype equals the friction factor in the model. ... 

The flow in fluid-fluid microstructured channels is characterized using dimensionless numbers. The most important dimensionless number for characterization of all types of flows is the Re number that relates inertial force to viscous force. Due to low flow velocities and characteristic dimension in the micrometer range, Re is often less than 1 meaning that viscous force is dominant over inertial force. The capillary number Ca is the ratio of viscous to interfacial forces. The range of Ca in a typical microchannel is lO " to 10 . Multiplying both numbers. Re and Ca, results in the Weber number We, which represents the ratio between inertial and interfacial forces. The importance of gravity vhth respect to interfacial forces is characterized by the Bond number Bo. The definitions of the dimensionless numbers are summarized in Table 2.2. 

If the equations above are made dimensionless there remain two important dimensionless numbers that govern the heat balances in the extruder the Graez number and the Brinkmann number. 

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